where \(\chi (M)\)  denotes the Euler number of \(M\)  and \({\rm deg}~D=\sum _{n=1}^{N}(\alpha _{n}-1)\)  the degree of the divisor \(D\) . If \(K\le 0\) , the metric exists if and only if \(\chi (M)+{\rm deg}~D\le 0\)  (see [1]},[2]}). If \(K> 0\) , the problem is still open now, except that there are some partial results. Troyanov [3]} proved that there exists a CSC-1 metric on \(S^{2}_{\lbrace \alpha ,\beta \rbrace }\)  if and only if \(\alpha =\beta \) . He also found a sufficient condition in [2]} under which there exists a CSC-1 metric with finite conical singularities on a compact Riemann surface \(M\) . Provided that \(M=S^{2}\)  and all angles lie in \((0,2\pi )\) , Luo and Tian [5]} proved that the sufficient of Troyanov is also necessary and the metric is unique. Under some restrictive conditions, Chen and Li [6]} found some necessary conditions of the existence of CSC-1 metrics on a compact Riemann surface with finite conical singularities. Chen et al [7]} studied the developing map of a CSC-1 metric with finite conical singularities, and converted the existence of a kind of CSC-1 metric with conical singularities to the existence of some kind of meromorphic 1-form on \(M\) . For more results about CSC metrics with singularities, we refer the readers to [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]} and references cited in.
